\be
\item Verify formula (\ref{E:Sensitivity_A_adjusted_delta}).
\item Consider the LP
	\begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2+x_3  \\
		\mbox{ subject to} \notag\\
		x_1 -2x_2+3x_3 &\leq 4 \notag\\
		x_1+3x_2 -2x_3 &\leq 15  \notag\\
		2x_1+x_2 +x_3&\leq 10 \notag \\
		x_1,x_2,x_3 &\geq 0.\notag
		\end{align*}
    \be
    \item Solve this LP. One of the three decision variables in your solution should be nonbasic.
    \item By how much can the objective coefficient of $x_1$ increase or decrease and the set of basic variables in the solution be unchanged from that of original LP? For this range of coefficient values, what are the values of the decision variables and objective function? Repeat this process, separately, for each of the other objective coefficients.
    \item By how much can the bound in the first constraint increase or decrease and the set
    of basic variables in the solution be unchanged from that of the original LP? For this range of values, what are the values of all variables, both decision and slack, as well as the objective function? What is the corresponding shadow price? Repeat this process for the bound in the second constraint.
    \item Now consider the LP formed by changing the bounds on the second and third constraints:
	\begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2+x_3  \\
		\mbox{ subject to} \notag\\
		x_1 -2x_2+3x_3&\leq 4  \notag\\
		x_1+3x_2 -2x_3 &\leq 15+\delta_2  \notag\\
		2x_1+x_2 +x_3&\leq 10 + \delta_3 \notag \\
		x_1,x_2 &\geq 0.\notag
		\end{align*}
	Sketch the set of ordered pairs,  $(\delta_2,\delta_3)$, for which the set of basic variables in the solution is the same as that of the original LP. For points in this set, what are the values of the decision variables and the objective function in terms of $\delta_2$ and $\delta_3$?
    \item By how much can the coefficient of $x_1$ in the second constraint increase or decrease and the set
    of basic variables in the solution be unchanged from that of the original LP? For this range of values, what are the values of all variables, both decision and slack, as well as the objective function?
    \ee
\item Suppose that \LP currently produces its two fuels in a manner that optimizes
profits but is considering adding a mid-grade blend to its line of products. In terms of the LP, this change corresponds to introducing a new decision variable. Each gallon of
	mid-grade fuel requires two gallons of stock A and two and a half gallons of stock B and, hence, diverts resources away from producing the other fuel types. The company seeks to determine the smallest net profit per gallon of mid-grade blend that is needed to increase the company's overall profit from its current amount.
\be
\item The introduction of a third fuel type adds an extra column
to the original coefficient matrix. Determine the new coefficient matrix, $\tilde{A}$.
\item Suppose that $p_m$ denotes the profit per gallon of the mid-grade fuel type. Then
$\tilde{\vecc}=[4,3,p_m]$ records the profit per gallon of each of the three fuel types. If
$\vecy_0$ denotes the solution of the original dual LP, calculate
$\ds -tilde{\vecc}+ \vecy_0 \tilde{A}$ in terms of $p_m$.
\item Explain why $\ds -tilde{\vecc}+ \vecy_0 \tilde{A}$ records the decision variable coefficients
in the top row of the tableau that result if the same steps of the simplex algorithm
used to solve the original LP are applied to the new LP having the third decision variable.
\item Use the result from the previous question to determine the smallest
net profit per gallon of mid-grade blend that is needed to increase the company's overall profit from its current amount.
\ee
	\ee

